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Fundamentals of Logic Design Chap. 6

QUINE-McCLUSKEY METHOD

This chapter in the book includes:

Objectives

Study Guide

6.1 Determination of Prime Implicants

6.2 The Prime Implicant Chart

6.3 Petrick’s Method

6.4 Simplification of Incompletely Specified Functions

6.5 Simplification Using Map-Entered Variables

6.6 Conclusion

Programmed Exercises

Problems

CHAPTER 6

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Fundamentals of Logic Design Chap. 6

Objectives

Topics introduced in this chapter:

1. Find the prime implicants of a function by using the Quine-McCluskey method.

2. Define prime implicants and essential prime implicants

3. Given the prime implicants,

find the essential prime implicants and a minimum sum-of-products expression for a function,

using a prime implicant chart and using Petrick method

4. Minimize an incompletely specified function, using the Quine-McCluskey method

5. Find a minimum sum-of-products expression for a function,

using the method of map-entered variables

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Fundamentals of Logic Design Chap. 6

6.1 Determination of Prime Implicants

XXYXY '

0 1 1 0 1 0 1 0

''''

X Y X Y X

-1 0 1 1 1 0 1 0 1 0 1

'

''''

BCDADBCA

CABCDABCDAB

(the dash indicates a missing variable)

(will not combine)

(will not combine)

The minterms are represented in binary notation and combined using

The binary notation and its algebraic equivalent

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Fundamentals of Logic Design Chap. 6

)14,10,9,8,7,6,5,2,1,0(),,,( mdcbaf

1110 14

0111 7 3 group

1010 10

1001 9

0110 6 2 group

0101 5

1000 8

0010 2 1 group

0001 1

0000 0 0 group

6.1 Determination of Prime Implicants

the binary minterms are sorted into groups

Is repersented by the following list of minterms:

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Fundamentals of Logic Design Chap. 6

6.1 Determination of Prime Implicants

Determination of Prime Implicants (Table 6-1)

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Fundamentals of Logic Design Chap. 6

6.1 Determination of Prime Implicants

''''''''' cddbcbbcabdadcaf (1,5) (5,7) (6,7) (0,1,8,9) (0,2,8,10) (2,6,10,14)

'''' cdcbbdaf

Definition: Given a function F of n variables, a product term P is an implicants of F iff for every combination of values of the n variables for which P=1, F is also equal to 1.

Definition: A Prime implicants of a function F is a product term implicant which is no longer an implicant if any literal is deleted from it.

The function is equal to the sum of its prime implicants

Using the consensus theorem to eliminate redundant terms yields

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Fundamentals of Logic Design Chap. 6

6.2 The Prime Implicant Chart

Essential Prime Implicant :

bdacdcbf ''''

Remaining cover

Prime Implicant Chart (Table 6-2)

The resulting minimum sum of products is

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Fundamentals of Logic Design Chap. 6

6.2 The Prime Implicant Chart

bdacdcbf ''''

The resulting minimum sum of products is

The resulting chart (Table 6-3)

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Fundamentals of Logic Design Chap. 6

6.2 The Prime Implicant Chart

7) 6, 5, 2, 1, (0, mF

Example: cyclic prime implicants(two more X’s in every column in chart)

Derivation of prime implicants

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Fundamentals of Logic Design Chap. 6

6.2 The Prime Implicant Chart

acbcbaF '''

The resulting prime implicant chart (Table 6-4)

One solution:

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Fundamentals of Logic Design Chap. 6

6.2 The Prime Implicant Chart

abcbcaF '''

Again starting with the other prime implicant that covers column 0.

The resulting table (Table6-5)

Finish the solution and show that

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Fundamentals of Logic Design Chap. 6

6.3 Petrick’s Method

1))()()()()(( 656453423121 PPPPPPPPPPPPP

minterm0 minterm1

- A technique for determining all minimum SOP solution from a PI chart

Because we must cover all of the minterms,

the following function must be true:

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Fundamentals of Logic Design Chap. 6

63264326321

6532

PPPPPPPPPPP

PPPP

))((

))()((

643154326521541

63563243262141

635624321

PPPPPPPPPPPPPPP

PPPPPPPPPPPPPP

PPPPPPPPPP

632643154326521541 PPPPPPPPPPPPPPPPPPP

acbcbaF ''' .''' abcbcaF or

6.3 Petrick’s Method

- Reduce P to a minimum SOP

- Choose P1,P4,P5 or P2,P3,P6 for minimum solution

First, we multiply out, using (X+Y)(X+Z) = X+YZ and

the ordinary Distributive law

Use X+XY=X to eliminate redundant terms from P

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Fundamentals of Logic Design Chap. 6

6.4 Simplification of Incompletely Specified Functions

)15,10,1()13,11,9,7,3,2(),,,( dmDCBAFExample:

Don’t care terms are treated like required minterms…

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Fundamentals of Logic Design Chap. 6

6.4 Simplification of Incompletely Specified Functions

ADCDCBF '

)()()( 1513119151173111032 mmmmmmmmmmmmF

1 1111, ;1 1010, ;0 ,0001 FforFforFABCDfor

Don’t care columns are omitted when forming the PI chart…

Replace each term in the final expression for F

The don’t care terms in the original truth table for F

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Fundamentals of Logic Design Chap. 6

6.5 Simplification Using Map-Entered Variables

terms)caret don'(

),,,,,( 1511975320

mmFmEmEmmmmFEDCBAG

Using of Map-Entered Variables (Figure 6-1)

The map represents the 6-variable function

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Fundamentals of Logic Design Chap. 6

6.5 Simplification Using Map-Entered Variables

)(),,,( '''''' CABABCDDBCABCACBADCBAF

Use a 3-variable map to simplify the function:

Simplification Using a Map-Entered Variable (Figure 6-2)

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Fundamentals of Logic Design Chap. 6

6.5 Simplification Using Map-Entered Variables

BDACDCABACDCAF '''' )(

...22110 MSPMSPMSF

FADDEAACDBAG '''

From Figure 6-2(b),

Find a sum-of-products expression for F of the form

The resulting expression is a minimum sum of products for G(Fig. 6-1) :

MS0 : minimum sum obtained by P1=P2=…=0

MS1 : minimum sum obtained by P1=1, Pj=0(j≠ 1) and replacing

all ‘1’’s on the map with ‘don’t cares(X)’

MS2 :….